Rota-Baxter operators were first appeared in the work of Baxter in his study of the fluctuation theory [1]} and further developed by Rota in combinatorics [2]}. Subsequently, Cartier [3]} and Atkinson [4]} studied further properties of Rota-Baxter operators. In last twenty years, Rota-Baxter operators on associative algebras pay very much attention due to its connection with combinatorics of shuffle algebras [5]}, Yang-Baxter equations [6]}, dendriform algebras [6]}, [8]}, renormalizations in quantum field theory [9]}, multiple zeta values in number theory [10]} and splitting of algebraic operads [11]}. See [12]} for more details about Rota-Baxter operators on associative algebras. On the other hand, Rota-Baxter operators on Lie algebras was first considered by Kuperscmidt in the study of classical \(r\) -matrices [13]}. They are also connected with pre-Lie algebras [11]}, integrable systems [15]} and combinatorics of Lyndon-Shirshov words [16]}.
